1. Field of the Invention
This invention relates to a temperature-insensitive semiconductor laser, such as a wavelength-stabilized semiconductor laser and a semiconductor laser having a high characteristic temperature, which can be preferably employed as a light source in a wavelength division multiplexing optical communication system and the like.
2. Related Background Art
In general, the oscillation wavelength of a semiconductor laser (a laser diode (LD)) is likely to vary depending on its ambient temperature. When the temperature changes, spacings between atoms and magnitudes of lattice vibrations vary in semiconductor crystals, and hence, the energy bandgap and refractive index thereof change. For this reason, the oscillation wavelength of the LD changes. Generally, the refractive index decreases as the bandgap increases.
A waveguide-type Fabry-Perot LD has a large number of resonance modes (longitudinal modes) in its gain spectrum, and oscillates at a wavelength of the resonance modes that is the closest wavelength to a wavelength at the gain peak. This resonance wavelength is proportional to an effective refractive index of a light waveguide when the effective refractive index is approximately uniform over the entire waveguide, while proportional to an optical length of its entire cavity when the effective refractive index varies along the waveguide. The gain-peak wavelength changes depending on the shape of its energy band structure. As a result, the oscillation wavelength of the Fabry-Perot LD depends on both the energy band structure and the refractive index (or the optical length of the cavity).
On the other hand, the oscillation wavelength of a distributed feedback semiconductor laser (DFB-LD) is determined by a pitch of its built-in diffraction grating and an effective refractive index of its light waveguide, provided that reflectivities at its end facets are negligibly low. In other words, its oscillation wavelength is independent of a wavelength at its gain peak. Thus, this oscillation wavelength is influenced only by the refractive index since the physical pitch is fixed. Naturally, no oscillation occurs when a difference between the gain-peak wavelength and the resonance wavelength is too large to obtain the oscillation threshold gain at the resonance wavelength. It does not, however, mean that the oscillation wavelength is under the influence of the wavelength at the gain peak.
Optical-fiber commmunication recently has been used in subscriber systems as well, as in trunk line systems as the information capacity has increased. The semiconductor laser of such subscriber systems is often used as a light source in hostile environments, such as a place where the temperature greatly changes, in contrast to that of the trunk line systems. Further, the demand for using such a semiconductor laser without any temperature controller has increased, because it reduces the cost.
Wavelength division multiplexing (WDM) transmission exists as a method for increasing the information transmission capacity. In WDM transmission, multiplexed wavelength signals must be precisely stabilized. Therefore, there is a need for a semiconductor laser, whose wavelength is stable even in a hostile atmosphere, such that the WDM transmission can be widely employed even in subscriber systems. Japanese Patent Application Laid-Open No. 9(1997)-219561 discloses such a wavelength-stabilized semiconductor laser.
In a device of this disclosure, its active layer and light guiding layer are composed of material whose bandgap (i.e., its refractive index) remains unchanged even when the temperature changes. Accordingly, where crystals are to be formed without any large difference between their lattice constants such that no inelastic strain is introduced therein, a combination of crystal materials must be selected from a narrow range since the lattice matching needs to be approximately attained between those crystals. Further, since laser light extends further to its substrate and cladding layer, it is impossible to maintain the effective refractive index of its light waveguide at a constant value when the temperature changes.
It is an object of the present invention to provide a temperature-insensitive semiconductor laser wherein one kind or a small number of kinds of materials constitute a layer structure, which is introduced to stabilize its cavity length or its effective refractive index of its waveguide, such that a change in its oscillation wavelength is satisfactorily small irrespective of a fluctuation in the temperature of the semiconductor laser.
Another object of the present invention is to provide a semiconductor laser wherein an overflow of carriers is suppressed against a fluctuation in the temperature of the semiconductor laser and its oscillation threshold current is thus stabilized.
The present invention is generally directed to a semiconductor laser which includes a substrate, semiconductor layers which are formed on the substrate and define a cavity including a waveguide with an active region, and which include at least one semiconductor layer whose refractive-index temperature coefficient is set at a non-positive or minute value to achieve at least one function of stabilizing a wavelength of the semiconductor laser and suppressing an overflow of carriers from the active region, and a driving unit for causing electron-hole recombination in the active region.
There are three configurations of the present invention, based on the above fundamental construction.
In accordance with a first configuration of the present invention, there is provided a semiconductor laser wherein the semiconductor layers formed on the substrate include at least one semiconductor layer whose temperature coefficient of the refractive-index (also referred to as a refractive-index temperature coefficient in this specification) is set at a negative value to achieve at least one of the functions of approximately maintaining an optical length of the cavity or an effective refractive index of the waveguide at a constant value against a change in the temperature of the semiconductor laser and suppressing an oscillation threshold current injected into the active region by the driving unit, such as a current-injecting unit, against a change in the temperature of the semiconductor laser.
The principle of the first configuration will be described in detail below. When a semiconductor layer with a negative refractive-index temperature coefficient is arranged in a laser-light existing region of the laser, both the above functions can be achieved. In contrast, when a semiconductor layer with a negative refractive-index temperature coefficient is arranged in a region of the laser where no laser light exists, only the above function of suppressing the oscillation threshold current injected into the active region against a change in the temperature of the laser can be achieved.
More specifically, the following specific structures are possible in the first configuration. The semiconductor layers formed on the substrate may include in the laser-light existing region a first semiconductor layer whose refractive-index temperature coefficient is set at a positive value and a second semiconductor layer whose refractive-index temperature coefficient is. set at a negative value to achieve the function of approximately maintaining the optical length of the cavity or the effective refractive index of the waveguide at a constant value against a change in the temperature of the laser. In this structure, the cavity length or the effective refractive index of the waveguide can be maintained approximately unchanged, and the oscillation wavelength can be hence stabilized.
The first semiconductor layer and the second semiconductor layer may be placed in an approximately parallel manner along a growth direction in the laser-light existing region. This relates to a structure wherein a compensation layer is disposed along the growth direction. The compensation layer will be described below in the description of the principle. This structure can be readily fabricated by an ordinary semiconductor growth process.
The waveguide may include a cladding layer, a light guiding layer and an active layer constituting the active region, and the first semiconductor layer and the second semiconductor layer may be any ones of the cladding layer, the light guiding layer and the active layer, respectively.
Either the first semiconductor layer or the second semiconductor layer may be the active layer. The active layer necessarily has a certain degree of an optical confinement factor, and therefore, this design makes it easy to stabilize the cavity length or the effective refractive index of the waveguide against any temperature change. In this case, the other one of the first semiconductor layer and the second semiconductor layer may be preferably placed under and/or above the active layer in parallel with the active layer.
A first semiconductor layer having a positive refractive-index temperature coefficient may be the active layer. In this structure, crystal quality of the active layer can be readily provided, leading to improved characteristics of the laser. In this case, a second semiconductor layer having a negative refractive-index temperature coefficient is arranged near the active layer. Therefore, the structure can readily achieve both of the above-discussed functions.
The semiconductor layers formed on the substrate may include in the laser-light existing region the first semiconductor layer and the second semiconductor layer formed on an approximately common plane perpendicular to the growth direction and along a light propagation direction or cavity-axial direction. This relates to a structure wherein the compensation layer is disposed on a lateral side of the active region. In this case, a plurality of the first semiconductor layers and a plurality of the second semiconductor layers may be alternately formed on the common plane along the light propagation direction.
Also in the structure right above, one of the first semiconductor layer and the second semiconductor layer may be the active layer. In this structure, the other one of the first semiconductor layer and the second semiconductor layer may be formed on at least one of the lateral opposite sides of the active layer on the common plane along the light propagation direction. The first semiconductor layer having a positive refractive-index temperature coefficient may be the active layer. Technical advantages of such a structure are described above.
A combination of the above layer arrangements in the growth direction and in the lateral direction may be adopted. In such a structure, the effective refractive index of the waveguide can be stabilized in both the growth and lateral directions. Therefore, the lasing wavelength can be further stabilized and the fluctuation in the oscillation threshold current can be further suppressed.
A relation of xcex2=xe2x88x92(1xe2x88x92xcex93)xcex1/xcex93 may be satisfied where xcex1 is the positive refractive-index temperature coefficient of the first semiconductor layer, xcex2 is the negative refractive-index temperature coefficient of the second semiconductor layer, and xcex93 is the optical confinement factor for the second semiconductor layer. This structure can further improve the stability of the oscillation wavelength.
The above structures may be adopted in a Fabry-Perot semiconductor laser, a distributed feedback semiconductor laser which includes a diffraction grating formed in the waveguide extending along the light propagation direction, or a distributed Bragg reflector semiconductor laser which includes a diffraction grating formed in a portion of the waveguide without the active region extending along the light propagation direction, which has a diffraction-grating region with the diffraction grating and an active-layer region with the active layer, and in which both the diffraction-grating region and the active-layer region include at least one semiconductor layer whose refractive-index temperature coefficient is negative, respectively.
The operation principle of the first configuration of the present invention is as follows.
Initially, the cavity length or the effective refractive index of the waveguide will be considered with respect to the stabilization of the oscillation wavelength. For simplicity, the refractive index of the semiconductor layers constituting the semiconductor laser is assumed to be n, and the compensation layer for compensating for a change in the refractive index is introduced. Layers with positive and negative temperature coefficients of refractive indices are those which mutually cancel out changes in their refractive indices, and one layer is accordingly named a compensation layer for the other layer. The effective refractive index neff of the waveguide is given by:
neff=(1xe2x88x92xcex93)ns+xcex93nxe2x80x83xe2x80x83(1),
where n is the refractive index of the compensation layer and xcex93 is the optical confinement factor for the compensation layer which is determined by a ratio of a portion of a light intensity distribution involved in the compensation layer relative to the entire light intensity distribution. In the case of the Fabry-Perot LD, the optical length of the cavity can be obtained by a product of neff and its physical cavity length. In the case of the DBR-LD, the optical length of its active-layer region can be obtained by a product of neff and the physical length of the active-layer region.
Since the resonance wavelength is proportional to the effective refractive index or the cavity length, the oscillation wavelength is stabilized when the effective refractive index neff is kept unchanged.
A condition therefor will be considered. Where xcex1 and xcex2 are temperature coefficients of refractive indices of the semiconductor layers and the compensation layer, respectively, and ns0 and n0 are refractive indices of the semiconductor layers and the compensation layer at a reference temperature respectively, dependencies of refractive indices of the semiconductor layers and the compensation layer on a temperature change xcex94T are represented as follows:
Ns=ns0+xcex1xcex94Txe2x80x83xe2x80x83(2),
and
n=n0+xcex2xcex94Txe2x80x83xe2x80x83(3).
By substituting relation (1) into relations (2) and (3), the effective refractive index neff of the waveguide is given by:
neff=[(1xe2x88x92xcex93)ns0+xcex93n0]+[(1xe2x88x92xcex93)xcex1+xcex93xcex2]xcex94Txe2x80x83xe2x80x83(4).
Since the first term on a right side in relation (4) is constant, the following condition is only required to maintain the effective refractive index neff at a constant value and stabilize the oscillation wavelength of the laser:
xcex2=xe2x88x92(1xe2x88x92xcex93)xcex1/xcex93xe2x80x83xe2x80x83(5).
More moderately, the laser only needs to be designed such that (1xe2x88x92xcex93)xcex1+xcex93xcex2≈0. The absolute value of the temperature coefficient xcex1 of semiconductor material used in the LD is ordinarily of the order of 10xe2x88x924Kxe2x88x921. The optical confinement factor xcex93 only needs to be set at 0.5 if such material as satisfies xcex2=xe2x88x92xcex1 exists, for example.
Next, the suppression effect of the fluctuation in the oscillation threshold current due to the semiconductor layer with a non-positive refractive-index temperature coefficient will be described.
The absorption coefficient xcex1 (E) of a semiconductor is approximately given for light having energy E above a bandgap Eg by the following relation:
xe2x80x83xcex1(E)≈2xc3x97104(Exe2x88x92Eg)cmxe2x88x921xe2x80x83xe2x80x83(6),
where E and Eg are expressed in units of an electron volts (eV).
From Kramers-Kronig relations, the refractive index n(E) is given by:
n(E)=1+cxc2x7h/(2xcfx802)xc2x7∫xcex1(Exe2x80x2)/(Exe2x80x2 2xe2x88x92E2)xc2x7dExe2x80x2
(the integration range is from 0 to ∞)
=1+cxc2x7h/(2xcfx802)xc2x7∫xcex1(Exe2x80x2)/(Exe2x80x2 2xe2x88x92E2)xc2x7dExe2x80x2
(the integration range is from Eg to ∞)
=1+cxc2x7h/(2xcfx80E)xc2x7104(E+Eg)xc2xdxe2x80x83xe2x80x83(7)
(relation (6) is used)
where c is the velocity of light in vacuum and h is Planck""s constant. Herein, the effect of band tailing is neglected since no substantial influence occurs to the consideration.
From relation (7), when E=Eg,
n(E)=1+cxc2x7h/(2xcfx80)xc2x7104(2/Eg)xc2xdxe2x80x83xe2x80x83(8).
Therefore, when Eg=0.8 eV (its corresponding wavelength xcexg=1.55 xcexcm) and both sides of relation (8) are differentiated with respect to temperature T, the following relation is obtained:
dn(Eg)/dT=xe2x88x920.2dEg/dTxe2x80x83xe2x80x83(9).
From relation (9), it is found that dEg/dTxe2x89xa70 in the material with a refractive-index temperature coefficient dn/dTxe2x89xa60. Therefore, a band offset (a difference in energy gap between adjacent layers) at the surface of the layer having a non-positive refractive-index temperature coefficient increases as the temperature rises. Even when dn/dT=0 (zero refractive-index temperature coefficient), the band offset at the layer surface increases as the temperature rises if its adjacent layer has a positive refractive-index temperature coefficient (its energy gap decreases as the temperature rises). Hence, the overflow of carriers can be suppressed against an increase in temperature (i.e., carriers become difficult to overflow as the band offset is enlarged), and the fluctuation in the oscillation threshold current with the temperature change can be reduced.
When the second semiconductor layer with a negative temperature coefficient is the compensation layer placed in the laser-light existing region, there can be provided a semiconductor laser in which not only fluctuation in the oscillation wavelength but fluctuation in the oscillation threshold current are greatly lowered, i.e., which has an excellent temperature characteristic.
The operation principle is the same as above in a semiconductor laser wherein the semiconductor layers formed on the substrate include, in the laser-light existing region, the first semiconductor layer and the second semiconductor layer formed on the approximately common plane perpendicular to the growth direction and along the light propagation direction or cavity-axial direction. The laser light is also distributed in the direction perpendicular to the growth direction. Therefore, when the optical confinement factor in the direction perpendicular to the growth direction is considered, the above temperature-insensitive semiconductor laser which has an excellent temperature characteristic as well as the stabilized oscillation wavelength can be realized based on the same principle as discussed above.
In accordance with a second configuration of the present invention, there is provided a semiconductor laser wherein the waveguide includes a first region with at least one first semiconductor layer having a positive refractive-index temperature coefficient and a second region with at least one second semiconductor layer having a negative refractive-index temperature coefficient to achieve the function of approximately maintaining the optical length of the cavity or the effective refractive index of the waveguide at a constant value against a temperature change of the semiconductor laser, and wherein the first region and the second region are arranged serially along the light propagation direction.
More specifically, the following specific structures are possible in the second configuration.
The waveguide may include a first region and a second region to achieve the function of approximately maintaining the optical length of the cavity against a temperature change of the semiconductor laser. This specific structure is described in a seventh embodiment described below.
The waveguide may include a plurality of the first regions and a plurality of the second regions arranged alternately and serially along the light propagation direction to achieve the function of approximately maintaining the length of the cavity or the effective refractive index of the waveguide at a constant value against a change in temperature of the semiconductor laser. This specific structure is described below in an eighth embodiment.
The first semiconductor layer and the second semiconductor layer may be any layers constituting the semiconductor laser structure, respectively, so long as the optical length of the cavity or the effective refractive index of the waveguide is maintained constant against a temperature change. Therefore, the first semiconductor layer and the second semiconductor layer may be any one of a cladding layer, a light guiding layer, and an active layer constituting the active region, respectively.
Typically, one of the first semiconductor layer and the second semiconductor layer is the active layer. In this case, a first semiconductor layer having a positive refractive-index temperature coefficient is preferably the active layer since the crystal quality of a semiconductor with a negative refractive-index temperature coefficient is likely to be deteriorated.
The above structures may be adopted in a Fabry-Perot semiconductor laser, a distributed feedback semiconductor laser which includes a diffraction grating formed in the waveguide extending along the light propagation direction, or a distributed Bragg reflector semiconductor laser which includes a diffraction grating formed in a portion of the waveguide without the active region extending along the light propagation direction, which has a diffraction-grating region with the diffraction grating and an active-layer region with the active layer, and in which both the diffraction-grating region and the active-layer region or only the diffraction-grating region includes the first region and the second region arranged serially along the light propagation direction.
The operation principle of the second configuration of the present invention is as follows.
In the above structure, the resonance wavelength is proportional to the optical length of the cavity (in the case of the Fabry-Perot LD), or the effective refractive index of the waveguide (in the case of the DFB-LD). In the case of the DBR-LD, the resonance wavelength is concerned with the effective refractive index of the diffraction-grating region and the optical length of the active-layer region. A condition that the optical length Leff or the effective refractive index is maintained constant is required to stabilize the oscillation wavelength. The condition for a constant optical length Leff is treated in the following discussion, but conditions for a constant optical length and for a constant effective refractive index are equivalent to each other if the effective refractive index is substantially uniform over the entire cavity. As a representative example, a laser structure with serially coupled first and second regions having positive and negative refractive-index temperature coefficients, respectively, is adopted for consideration.
Where np and nn are effective refractive indices of the first and second regions, respectively, Lp and Ln are physical lengths of the first and second regions, respectively, xcex1 and xcex2 are temperature coefficients of refractive indices of the first and second regions, respectively, and np0 and nn0 are refractive indices of the first and second regions at the reference temperature, respectively, dependencies of the refractive indices on a change in temperature xcex94T in the first and second regions are given by:
np=np0+xcex1xcex94Txe2x80x83xe2x80x83(10),
and
nn=nn0+xcex2xcex94Txe2x80x83xe2x80x83(11).
When relations (10) and (11) are used, the condition for a constant optical length Leff of the optical cavity is given as follows:
Leff=npLp+nnLn=[np0Lp+nn0Ln]+[xcex1Lp+xcex2Ln]xcex94Txe2x80x83xe2x80x83(12).
Since the first term on a right side of relation (12) is constant, the following condition is only required to maintain the optical length Leff of the optical cavity at a constant value:
xcex2=xe2x88x92Lp/Lnxc2x7xcex1xe2x80x83xe2x80x83(13).
The absolute value of the temperature coefficient a of semiconductor material used in the LD is of the order of 10xe2x88x924Kxe2x88x921 in almost all cases. A relation of Lp=Ln is only required if such material that satisfies xcex2=xe2x88x92xcex1 exists, for example.
In the above description of the principle, no description is made to the active layer, but the gain spectrum slightly varies with the temperature change in a Fabry-Perot LD wherein the refractive-index temperature coefficient of the active layer is not zero. Therefore, its wavelength stability is a little poorer than that of the other types. When the active layer is formed of a semiconductor material whose refractive-index temperature coefficient is zero, that problem of poorer wavelength stability can be solved.
Further, the effective refractive index of the waveguide is not referred to in the above description of the principle of the second configuration, except special cases. Therefore, the wavelength stability cannot be sufficiently secured in DFB-LD and DBR-LD. However, the wavelength stability can also be obtained in those two LDs, when the first and second regions are alternately arranged as in the second embodiment later described and the effective refractive index is hence made substantially uniform over the entire waveguide, for example. In the DFB-LD, even where the optical length is made constant as discussed in the description of the principle, a certain degree of wavelength stability can be obtained if the diffraction grating is designed such that phases of light coincide with each other after one-round trip of the light. Furthermore, when the optical length is made constant in both the diffraction-grating region and the active-layer region of the DBR-LD, a certain degree of the wavelength stability can also be obtained. Even where the optical length is made constant only in the diffraction-grating region of the DBR-LD, a certain degree of the wavelength stability can be obtained if the diffraction grating is designed such that phases of light coincide with each other after one-round trip of the light.
In accordance with a third configuration of the present invention, there is provided a semiconductor laser wherein absolute values of refractive-index temperature coefficients of the semiconductor layers in the waveguide in a laser-light existing region are all set to be below 10xe2x88x925Kxe2x88x921 for a wavelength of laser light radiated in the active region. In this structure, the dependence of a change in the refractive index of the waveguide on a change in the temperature is made approximately equal to that of a dielectric light waveguide, for example. Therefore, light from the semiconductor laser can be smoothly guided into the dielectric light waveguide without any reflection when an integrated optical circuit is fabricated using those semiconductor laser and dielectric waveguide.
In this structure, where ns and xcex93s are refractive index and optical confinement factor for each semiconductor layer constituting the semiconductor laser for the convenience of simplicity, respectively, the effective refractive index neff of the waveguide in the laser-light existing region is approximately given by (the effectiveness of the following discussion is not lost by this assumption):
neff=xcexa3sxcex93snsxe2x80x83xe2x80x83(14)
(xcexa3s means the summation with respect to s).
Here, when a refractive-index temperature coefficient xcex1s is introduced, then the dependence of the refractive index ns on a temperature change xcex94T is represented by the following relation:
ns=ns0+xcex1sxcex94Txe2x80x83xe2x80x83(15).
Substituting relation (15) into relation (14), the effective refractive index neff is given by:
xe2x80x83neff=xcexa3sxcex93s[ns0+xcex1sxcex94T]xe2x80x83xe2x80x83(16).
In relation (16), since the first term in brackets is constant, the effective refractive index neff is necessarily made constant in a region of xcex93sxe2x89xa00 when the following relation is satisfied:
xcex1s=0xe2x80x83xe2x80x83(17).
In other words, the resonance wavelength is approximately maintained at a constant value against the temperature change when the refractive-index temperature coefficient is approximately zero in the semiconductor layer placed in the laser-light existing region (i.e., the semiconductor layer having a non-zero optical confinement factor).
The refractive index of a semiconductor layer placed in a region of xcex93s=0 (i.e., a region where no laser light exists) does not contribute to the effective refractive index, and therefore, the refractive-index temperature coefficient of this semiconductor layer need not be zero.
Ideally, it is preferable that all refractive-index temperature coefficients xcex1s are all zero. However, though not zero, a fluctuation in the oscillation wavelength with the temperature change can be lowered when its value is small. The refractive-index temperature coefficient of semiconductor material of an ordinary semiconductor laser is as xcex1s≈10xe2x88x924Kxe2x88x921, and the temperature coefficient of the dielectric light waveguide of glass or the like is about a tenth ({fraction (1/10)}) thereof. Therefore, when the semiconductor laser using a semiconductor layer with xcex1s=10xe2x88x925Kxe2x88x921 and the dielectric light waveguide are integrated to fabricate the optical circuit, both the laser and the dielectric waveguide exhibit approximately the same change in the refractive index such that light is smoothly propagated therebetween. Where the absolute value of xcex1s is further reduced, the fluctuation in the wavelength with the temperature change can be naturally further reduced.
More specifically, the following specific structures are possible in the third configuration.
Naturally, all the absolute values of the temperature coefficients may be set at zero for the wavelength of laser light radiated in the active region. In this case, it is found from the above relation (9) that dEg/dT=0 is established in the material with a zero refractive-index temperature coefficient (dns/dT=0). As a result, the bandgap also can be approximately maintained at a constant value even when the temperature changes. Therefore, temperature-insensitive Fabry-Perot, DFB and DBR semiconductor lasers can be realized. Especially, the oscillation wavelength stability of the Fabry-Perot LD is prominent since its oscillation wavelength is influenced by both the energy band and the effective refractive index.
The cavity may include at least one semiconductor layer with a negative refractive-index temperature coefficient arranged in a region where no laser light exists. In this structure, fluctuations in both the oscillation wavelength and the oscillation threshold current can be greatly lowered.
The semiconductor layer with a negative refractive-index temperature coefficient may be placed in parallel with the active layer. In this case, the semiconductor layer with a negative refractive-index temperature coefficient may be placed under and/or above the active layer in parallel with the active layer.
The above structures may be adopted in a Fabry-Perot semiconductor laser, a distributed feedback semiconductor laser which includes a diffraction grating formed in the waveguide extending along the light propagation direction, or a distributed Bragg reflector semiconductor laser which includes a diffraction grating formed in a portion of the waveguide without the active region extending along the light propagation direction, which has a diffraction-grating region with the diffraction grating and an active-layer region with the active layer, and in which both the diffraction-grating region and the active-layer region include the above semiconductor-layer structures. The active-layer region may include a semiconductor layer with a negative refractive-index temperature coefficient.
These advantages and others will be more readily understood in connection with the following detailed description of the more preferred embodiments in conjunction with the drawings.